How accurate is Newtonian Gravity? I know that really fast moving things need Relativity rather than Newtonian physics.
I also know the quirk of the Mercury´s orbit.
But just how much more accurate is General Relativity than Newton´s Law of Gravitation for predicting say the orbit of Earth or Neptune?
Can the "slingshot" effect where we use another planet´s gravity to accelerate a space probe be done with Newton or does that require General Relativity?
Is the speed of Jupiter (18 km/s I think) fast enough to make a difference in the accuracy of GR v Newton´s Law of Gravity?
 A: There are several different questions embedded in here.  The answer to all of them is "Newton's theory is accurate to great precision, and beyond measurement accuracy for most of your examples".  
The key point is that Newtonian physics fails when, roughly, the quantity $v/c > .1$ or $\frac{GM}{c^{2}r} > .1$.  You can calculate both of these quantities for the cases of Earth and Jupiter, and you will find that your answer is quite small.
A: The anomalous perihelion shift of Earth is 3.84 arc-seconds per century, or about one tenth the size of Mercury's shift. The anomalous perihelion shift of Jupiter is 0.0622 arc-seconds per century, or about one thousandth the size of Mercury's shift.
I can't find calculations for the effect of GR on slingshots, but I believe it to be negligable. The only significant deviation that has been found is the flyby anomaly, and GR does not explain this.
A: Newtonian gravity is very accurate at all scales and all distances except for velocities close to the speed of light.
